The determination of free energies that govern protein-protein recognition is essential for a detailed molecular understanding of biological specificity. Continuum models of macromolecular interactions, in which the solvent is treated by an implicit representation and the proteins are treated semi-microscopically, are computationally trac-table for estimating free energies, yet many questions remain concerning their accuracy. This article reports a continuum analysis of the free-energy changes underlying the binding of 31 interfacial alanine substitutions of two complexes of the anti-hen egg white lysozyme (HEL) antibody D1.3 bound with HEL or the antibody E5.2. Two implicit schemes for modeling the effects of protein and solvent relaxation were examined, in which the protein environment was treated as either homogeneous with a ''protein dielectric constant'' of p 4 or inhomoge-neous, with p 4 for neutral residues and p 25 for ionized residues. The results showed that the non-uniform dielectric model reproduced the experimental differences better, with an average absolute error of 1.1 kcal/mol, compared with 1.4 kcal/mol for the uniform model. More importantly, the error for charged residues in the nonuniform model is 0.8 kcal/mol and is nearly half of that corresponding to the uniform model. Several substitutions were clearly problematic in determining qualitative trends and probably required explicit structural reorganization at the protein-protein interface.
The emergence of the COVID-19 pandemic has brought increased attention to the critical mathematical literacy of citizens in the United States and around the world. A statistically and mathematically literate society is crucial for ensuring that citizens are able to sift through political rhetoric to maintain life-saving procedures such as social distancing and other infection dampening efforts. Additionally, recent civil unrest due to the disproportionate killings of Black men by police provokes investigation into the public's mathematical literacy. In this paper, we investigate adolescent students' critical mathematics consciousness and mathematics literacy as they reason through two interview tasks on the coronavirus and police shooting data. Drawing on Frankenstein's program of Critical Mathematics Education, we introduce an analytic framework for documenting the critical mathematics consciousness of adolescent students. We interviewed fifteen 14-to 16-yearold students as they solved five tasks designed to elicit their critical and ethical mathematical awareness. Our findings indicate that students exhibit very little critical mathematics consciousness in the context of the police problem but show awareness that data can be presented in ways that manipulate the public's emotions in the coronavirus problem. We conclude the paper with a discussion of implications for designing future instruction to support students' growth in critical mathematics consciousness.
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